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Bulletin of the American Physical Society

Parallel quantum Monte Carlo: The one-dimensional Kondo insulator

Richard M. Fye, Douglas J. Scalapino

Published May 1, 1993• 0 citations
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Abstract

The authors discuss parallelization strategies for the BSS (Blankenbecler-Sugar-Scalapino) algorithm, a quantum Monte Carlo technique for treating lattice fermions. They then present results obtained by applying the BSS algorithm to the one-dimensional Kondo insulator model on both vector and parallel machines. In particular they compare Kondo insulator spin correlations with RKKY predictions, investigate the opening of a charge gap, and consider the question of long-range anti-ferromagnetic order.

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The authors discuss parallelization strategies for the BSS (Blankenbecler-Sugar-Scalapino) algorithm, a quantum Monte Carlo technique for treating lattice fermions. They then present results obtained by applying the BSS algorithm to the one-dimensional Kondo insulator model on both vector and parallel machines. In particular they compare Kondo insulator spin correlations with RKKY predictions, investigate the opening of a charge gap, and consider the question of long-range anti-ferromagnetic order.
This thesis discusses mainly two Fermionic lattice systems, first a Kondo lattice with additional Hubbard interaction and second a Hubbard Hamiltonian augmented with additional spin and charge interactions. We first introduce the Quantum Monte Carlo technique, which is then employed to study the two respective systems. We present an innovation that allows to calculate time displaced Greens functions more efficiently. Compared with previously used numerically stable algorithms the new method gains an order of magnitude in speed, but is just as precise, and very simple to implement. In the second chapter we consider the Kondo lattice model in two dimensions at half filling. In addition to the Fermionic hopping integral t and the superexchange coupling J the role of a Coulomb repulsion U in the conduction band is investigated. We find the model to display a magnetic order-disorder transition in the U-J plane with a critical value of Jc which is decreasing as a function of U. The single-particle spectral function A(k,ω) is computed across this transition. We conclude that (i) the local screening of impurity spins determines the low-energy behavior of the spectral function and (ii) one cannot deform continuously the spectral function of the half-filled Hubbard model at J=0 to that of the Kondo insulator at J>Jc. In the third chapter we investigate the phase diagram of a new model that exhibits a first order transition between s-wave superconducting and antiferromagnetic phases. The model, a generalized Hubbard model augmented with competing spin-spin and pair-pair interactions, was investigated using the projector quantum Monte Carlo method. Upon varying the Hubbard U from attractive to repulsive, we find a first order phase transition between superconducting and antiferromagnetic states.
The 2D half-filled Kondo lattice model with exchange JJJ and nearest neighbor hopping ttt is considered. It is shown that this model belongs to a class of Hamiltonians for which zero-temperature auxiliary field Monte Carlo methods may be efficiently applied. We compute the staggered moment and spin and quasiparticle gaps on lattice sizes up to $12\ifmmode\times\else\texttimes\fi{}12$. The competition between the RKKY interaction and Kondo effect leads to a continuous quantum phase transition between antiferromagnetic and spin-gapped insulators. This transition occurs at $J/t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.45\ifmmode\pm\else\textpm\fi{}0.05$.
In order to better understand Kondo insulators, we have studied both the symmetric and asymmetric Anderson lattices at half filling in one dimension using the density-matrix formulation of the numerical renormalization group. The asymmetric case is treated in the mixed-valence regime. We have calculated the charge gap, the spin gap, and the quasiparticle gap as a function of the repulsive interaction U using open boundary conditions for lattices as large as 24 sites. We find that the charge gap is larger than the spin gap for all U for both the symmetric and asymmetric cases. Ruderman-Kittel-Kasuya-Yosida interactions are evident in the f-spin--f-spin correlation functions at large U in the symmetric case, but are suppressed in the asymmetric case as the f level approaches the Fermi energy. This suppression can also be seen in the staggered susceptibility \ensuremath{\chi}(q=2kF{\mathit{k}}_{\mathit{F}}kF​) and it is consistent with neutron scattering measurements of \ensuremath{\chi}(q) in CeNiSn.
We have studied the one-dimensional Kondo chain at half filling using a density matrix formulation of the numerical renormalization group. The charge gap is larger than the spin gap for all antiferromagnetic values of the exhange coupling J. A new type of excitation, a neutral spin singlet, consists of a particle and a hole which are repulsive for J\ensuremath{\gtrsim}5t and attractive for J\ensuremath{\lesssim}5t. As J\ensuremath{\rightarrow}0, RKKY interactions become more important and the staggered susceptibility \ensuremath{\chi}(q=2kF{\mathit{k}}_{\mathit{F}}kF​) diverges. We have also studied \ensuremath{\chi}(q) and the dispersion of the low-lying spin excitations as a function of wave vector q.
Ground-state properties of the one-dimensional Kondo lattice model are investigated using the quantum Monte Carlo (QMC) simulation method. To reduce the difficulty of the negative sign problem, we employ the variationally optimized singlet wave function. Since the improvement of Monte Carlo sampling efficiency is achieved in the half-filling case, we can perform the simulations on large systems (up to 26-site system). The QMC data on the local spin correlations are closely compared with those of the variational Monte Carlo (VMC) calculations as well as with the exact diagonalization data of the Ruderman-Kittel-Kasuya-Yoshida (RKKY) lattice model. We find that the local spin correlation decays exponentially with the correlation length ${\ensuremath{\xi}}_{\mathit{f}\mathit{f}}$\ensuremath{\sim}2.54 in the QMC data, while ${\ensuremath{\xi}}_{\mathit{f}\mathit{f}}$\ensuremath{\sim}1.08 in the VMC results (J/t=1). This implies the absence of the long-range order (the existence of the spin gap) and the importance of intersite correlation effects which are not taken into account in the VMC results. On the other hand, the system size dependence is almost absent in the Kondo lattice model results, which contrasts the behavior in the RKKY data.
A comparative study of the efficiency of different algorithms (Metropolis, Langevin, and Hybrid) for the simulation of a one-dimensional spinless fermion model with a half-filled band is presented. The Metropolis updating of the fields is performed with an algorithm introduced by Blankenbecler, Scalapino, and Sugar. The density-density correlation function, the order parameter, and the autocorrelation are calculated. A study of the dependence of the systematic errors on the temperature and the step size for the Langevin and hybrid algorithms is presented. The relation between the accuracy and the speed of the algorithms is studied.